An Ordinal Symptom-Scale PK/PD Case

Why Model the Full Symptom Scale?

A symptom rating often records ordered severity categories, such as none, mild, moderate, and severe. Reducing those ratings to “responder” or “nonresponder” can discard information: it treats a one-category improvement and a three-category improvement as equivalent, and the chosen cutoff determines which changes count. An ordinal exposure-response model instead uses the ordering of all categories while estimating how exposure shifts the probability of lower symptom severity.

Repeated ratings from one patient are also related. A patient who tends to report more severe symptoms may do so at several visits, even after accounting for measured exposure. This case uses a cumulative-logit proportional-odds model with a patient-specific random intercept to account for that repeated-measures structure.

Definition: Let $Y_{ij}$ be the ordinal symptom rating for patient $i$ at visit $j$, with categories $0,1,\ldots,K$ ordered from least to most severe. Let $C_{ij}$ be the exposure at that visit and define $x_{ij}=\log_2(C_{ij}/C_{\mathrm{ref}})$, where $C_{\mathrm{ref}}$ is a reference concentration. A random-intercept proportional-odds model is $$ \operatorname{logit}\{P(Y_{ij}\leq k\mid x_{ij},b_i)\} =\alpha_k+\beta x_{ij}+b_i,\qquad k=0,\ldots,K-1, $$ where $\operatorname{logit}(p)=\log\{p/(1-p)\}$, the cutpoints satisfy $\alpha_0<\alpha_1<\cdots<\alpha_{K-1}$, and $b_i\sim N(0,\tau^2)$. The random intercept accounts for persistent patient-to-patient differences. With this coding, $\beta>0$ means that higher exposure is associated with a greater probability of being at or below any symptom threshold.

How the Model Produces Category Probabilities

The model specifies cumulative probabilities first. Write $F_k(x,b)=P(Y\leq k\mid x,b)$. For interior categories, the probability of category $k$ is the difference between adjacent cumulative probabilities:

P(Y=0\mid x,b)=F_0(x,b),\qquad P(Y=k\mid x,b)=F_k(x,b)-F_{k-1}(x,b),\qquad P(Y=K\mid x,b)=1-F_{K-1}(x,b).

These probabilities are valid when the cutpoints are ordered and the model uses the same slope at every threshold. The inverse-logit function is increasing, so ordered cutpoints imply $F_0\leq F_1\leq\cdots\leq F_{K-1}$; consequently, the differences cannot be negative and the category probabilities sum to one. This is a useful basic check on implementation and reported predictions.

The proportional-odds assumption is that exposure has the same slope $\beta$ for every cumulative split $Y\leq k$ versus $Y>k$. Therefore, increasing $x$ by one unit multiplies each such cumulative odds by $\exp(\beta)$. Since $x$ is measured in exposure doublings, the multiplier describes a doubling of concentration. This is not a claim that the probability of every individual category changes by the same amount.

Worked Example: Predicting All Four Symptom Categories

Suppose a hypothetical four-category scale is coded 0 = none, 1 = mild, 2 = moderate, and 3 = severe. A fitted model has cutpoints $\alpha_0=-1.20$, $\alpha_1=0.2007$, and $\alpha_2=1.50$, with $\beta=0.80$. Let $C_{\mathrm{ref}}=2$ mg/L. For a patient with $b_i=0$ at $C=2$ mg/L, $x=\log_2(2/2)=0$, so the three cumulative logits are $-1.20$, $0.2007$, and $1.50$.

Applying the inverse logit gives $F_0=1/(1+\exp(1.20))\approx0.231475$, $F_1=1/(1+\exp(-0.2007))\approx0.55000725$, and $F_2=1/(1+\exp(-1.50))\approx0.817574$. Subtract the unrounded cumulative probabilities to obtain the category predictions:

\begin{aligned} P(Y=0)&\approx0.231475,\\ P(Y=1)&\approx0.55000725-0.2314752165...=0.318532,\\ P(Y=2)&\approx0.81757448-0.55000725=0.267567,\\ P(Y=3)&\approx1-0.81757448=0.182426. \end{aligned}

The rounded probabilities sum to 1.000000. In this conditional prediction for a patient with random intercept zero, the most probable rating is mild. It is not correct to call the probability of a mild rating 0.550: that is the probability of a rating of mild or less.

Repeated Ratings and Model Assumptions

The example represents the conditional model at $b_i=0$, not an average over the patient population. In fitting, the random intercept is integrated over its assumed normal distribution. Conditional on the random intercept and modeled covariates, ratings are assumed independent across visits; marginally, the shared $b_i$ induces within-patient association. This is a compact way to allow each patient a persistent tendency toward lower or higher symptom severity.

The model also assumes that the ordered categories are used consistently across visits, that exposure is measured on a meaningful scale, and that the stated exposure-response form is adequate over the observed range. If visit timing or disease progression matters, it must be represented in the model rather than silently attributed to exposure. A time term, for example, can be included when justified by the design and scientific question.

In a typical analysis, parameters are estimated by maximum likelihood, with the random effects integrated out. A compact hypothetical output might report $\hat\beta=0.80$ and $SE=0.32$. The standard error describes the estimated uncertainty in the coefficient; it does not describe the variation in individual ratings. The following calculation gives a Wald test of $H_0:\beta=0$ against $H_a:\beta\ne0$.

Worked Example: Interpreting the Exposure Effect

Using the hypothetical estimate $\hat\beta=0.80$ and $SE=0.32$, the Wald statistic is

z=\frac{\hat\beta}{SE}=\frac{0.80}{0.32}=2.50.

Under $H_0$, the approximate two-sided p-value is $2P(Z\geq2.50)\approx0.0124$, where $Z\sim N(0,1)$. At $\alpha=0.05$, this is evidence against a zero exposure slope under the fitted model. It is not the probability that $H_0$ is true.

A 95% Wald interval for $\beta$ is $0.80\pm1.96(0.32)$, or $0.1728$ to $1.4272$ log cumulative-odds units per exposure doubling. Exponentiating gives $\exp(0.1728)\approx1.189$ to $\exp(1.4272)\approx4.167$. The estimated cumulative odds ratio per doubling is $\exp(0.80)\approx2.226$, with a 95% interval of approximately 1.189 to 4.167. Thus, conditional on the model and patient random intercept, a doubling of exposure is associated with estimated 2.226-fold cumulative odds of being at or below each symptom threshold. This interval is wide, so the magnitude remains uncertain.

Translating Improvement Into an Exposure Change

A coefficient can also be translated into the exposure change associated with a specified clinically relevant probability shift. For a particular cumulative threshold $k$, let $p_b$ be the probability of $Y\leq k$ at a baseline exposure and $p_t$ the target probability at a higher exposure. Subtracting the two cumulative-logit equations cancels the cutpoint and random intercept:

\operatorname{logit}(p_t)-\operatorname{logit}(p_b) =\beta(x_t-x_b).

Because $x=\log_2(C/C_{\mathrm{ref}})$, the exposure ratio is $C_t/C_b=2^{x_t-x_b}$. This calculation describes a model-implied change, not proof that increasing exposure will cause the improvement. It also does not establish that the target exposure is safe or attainable.

Worked Example: Exposure Change for a Clinically Relevant Threshold Shift

Suppose the clinically relevant improvement is a rise from a 50% to a 75% probability of a rating of mild or less, $Y\leq1$, and use the hypothetical $\hat\beta=0.80$. The logit difference is

\operatorname{logit}(0.75)-\operatorname{logit}(0.50) =1.098612-0=1.098612.

The required change in log-base-2 exposure is $1.098612/0.80=1.373265$ doublings. Therefore, the estimated exposure ratio is $2^{1.373265}\approx2.5906$, or approximately 2.591. Under this model, moving from the baseline concentration to about 2.591 times that concentration corresponds to the stated change in probability. The estimate is conditional on the proportional-odds model, and uncertainty in $\beta$ makes the exposure ratio uncertain as well.

Checking Proportional Odds and Category-Level Fit

A plausible coefficient and a significant test do not establish that the model describes the data adequately. The central structural question is whether a common exposure slope is reasonable across the cumulative thresholds. Compare threshold-specific exposure patterns or fit a model allowing separate slopes as a diagnostic. Strongly different estimated slopes, systematic lack of fit at one threshold, or residual patterns related to exposure or visit can indicate that proportional odds is inadequate. A formal comparison should account for the models and estimation method used; a diagnostic test alone does not replace examination of the observed and predicted responses.

Predicted category probabilities should also be compared with observed category frequencies across useful exposure ranges and visits. Look for systematic underprediction of severe ratings, overprediction of middle categories, or deterioration at the extremes of exposure. For a random-intercept model, be explicit about whether predictions are conditional for a chosen $b_i$ or averaged over its distribution: those quantities need not be equal because the inverse-logit transformation is nonlinear.

A cumulative-logit model is not the only option. A binary responder model is simpler when a single threshold is the prespecified clinical endpoint, but it discards distinctions among the remaining categories. A multinomial model estimates category probabilities without using the ordinal structure and can avoid the common-slope restriction, at the cost of more parameters and less direct use of ordering. Other ordinal models, including adjacent-category or continuation-ratio models, answer differently structured questions. The choice should reflect how categories are generated and what clinical contrast matters, not merely which model produces the most favorable result.

Common Pitfalls and Limitations

Rounding cumulative probabilities before subtraction. This can make category probabilities inaccurate. Retain full precision through the differences, then round the final probabilities.
Calling a cumulative probability a category probability. $P(Y\leq1)$ includes both none and mild ratings; it is not $P(Y=1)$.
Ignoring repeated observations. Treating every visit as an independent patient can understate uncertainty and distort inference. Represent within-patient dependence, for example with a random effect or an appropriate marginal model.
Assuming a significant common slope validates proportional odds. Evidence that $\beta\ne0$ does not show that the same slope is suitable at every threshold. Check threshold-specific fit and predictions.
Interpreting an exposure ratio as a treatment recommendation. The model-based ratio does not account by itself for safety, exposure feasibility, confounding, or uncertainty in the full exposure-response relationship.

The model is limited by its assumptions about category ordering, cutpoints, dependence, and exposure response. Sparse categories can make threshold estimates unstable, and a random-intercept model does not automatically handle every source of time-varying confounding or informative missingness. Report the scale definition, exposure units and reference, repeated-measures structure, estimation approach, uncertainty, and category-level diagnostics alongside the coefficient.

Check Your Understanding

Use the model and hypothetical estimates in this tutorial. Solutions follow the questions.

If $\beta=0.80$, what is the estimated cumulative odds ratio for a one-unit increase in $x$, and what exposure change does one unit represent?
Why is $P(Y\leq1)$ not the probability of a mild rating alone?
Under the fitted proportional-odds model, what exposure ratio corresponds to increasing a cumulative probability from 0.50 to 0.75?
Name one diagnostic finding that could challenge the proportional-odds assumption.

Solutions. (1) The ratio is $\exp(0.80)\approx2.226$; one unit of $x=\log_2(C/C_{\mathrm{ref}})$ is a doubling of exposure. (2) The event $Y\leq1$ includes both none (0) and mild (1), so it combines two categories. (3) The logit difference is $1.098612$; dividing by $0.80$ gives 1.373265 doublings, and $2^{1.373265}\approx2.5906$, or about 2.591 times the baseline exposure. (4) For example, clearly different exposure slopes across cumulative thresholds, or systematic mismatch between observed and predicted category frequencies at a threshold, would challenge the assumption.